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## G = C5×D32order 320 = 26·5

### Direct product of C5 and D32

direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary

Aliases: C5×D32, C321C10, C1603C2, D161C10, C40.68D4, C20.39D8, C10.15D16, C80.19C22, C8.5(C5×D4), C4.1(C5×D8), (C5×D16)⋊5C2, C2.3(C5×D16), C16.2(C2×C10), SmallGroup(320,176)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C16 — C5×D32
 Chief series C1 — C2 — C4 — C8 — C16 — C80 — C5×D16 — C5×D32
 Lower central C1 — C2 — C4 — C8 — C16 — C5×D32
 Upper central C1 — C10 — C20 — C40 — C80 — C5×D32

Generators and relations for C5×D32
G = < a,b,c | a5=b32=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C5×D32
On 160 points
Generators in S160
(1 56 96 118 153)(2 57 65 119 154)(3 58 66 120 155)(4 59 67 121 156)(5 60 68 122 157)(6 61 69 123 158)(7 62 70 124 159)(8 63 71 125 160)(9 64 72 126 129)(10 33 73 127 130)(11 34 74 128 131)(12 35 75 97 132)(13 36 76 98 133)(14 37 77 99 134)(15 38 78 100 135)(16 39 79 101 136)(17 40 80 102 137)(18 41 81 103 138)(19 42 82 104 139)(20 43 83 105 140)(21 44 84 106 141)(22 45 85 107 142)(23 46 86 108 143)(24 47 87 109 144)(25 48 88 110 145)(26 49 89 111 146)(27 50 90 112 147)(28 51 91 113 148)(29 52 92 114 149)(30 53 93 115 150)(31 54 94 116 151)(32 55 95 117 152)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)(95 96)(97 106)(98 105)(99 104)(100 103)(101 102)(107 128)(108 127)(109 126)(110 125)(111 124)(112 123)(113 122)(114 121)(115 120)(116 119)(117 118)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)

G:=sub<Sym(160)| (1,56,96,118,153)(2,57,65,119,154)(3,58,66,120,155)(4,59,67,121,156)(5,60,68,122,157)(6,61,69,123,158)(7,62,70,124,159)(8,63,71,125,160)(9,64,72,126,129)(10,33,73,127,130)(11,34,74,128,131)(12,35,75,97,132)(13,36,76,98,133)(14,37,77,99,134)(15,38,78,100,135)(16,39,79,101,136)(17,40,80,102,137)(18,41,81,103,138)(19,42,82,104,139)(20,43,83,105,140)(21,44,84,106,141)(22,45,85,107,142)(23,46,86,108,143)(24,47,87,109,144)(25,48,88,110,145)(26,49,89,111,146)(27,50,90,112,147)(28,51,91,113,148)(29,52,92,114,149)(30,53,93,115,150)(31,54,94,116,151)(32,55,95,117,152), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)(95,96)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153)>;

G:=Group( (1,56,96,118,153)(2,57,65,119,154)(3,58,66,120,155)(4,59,67,121,156)(5,60,68,122,157)(6,61,69,123,158)(7,62,70,124,159)(8,63,71,125,160)(9,64,72,126,129)(10,33,73,127,130)(11,34,74,128,131)(12,35,75,97,132)(13,36,76,98,133)(14,37,77,99,134)(15,38,78,100,135)(16,39,79,101,136)(17,40,80,102,137)(18,41,81,103,138)(19,42,82,104,139)(20,43,83,105,140)(21,44,84,106,141)(22,45,85,107,142)(23,46,86,108,143)(24,47,87,109,144)(25,48,88,110,145)(26,49,89,111,146)(27,50,90,112,147)(28,51,91,113,148)(29,52,92,114,149)(30,53,93,115,150)(31,54,94,116,151)(32,55,95,117,152), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)(95,96)(97,106)(98,105)(99,104)(100,103)(101,102)(107,128)(108,127)(109,126)(110,125)(111,124)(112,123)(113,122)(114,121)(115,120)(116,119)(117,118)(129,144)(130,143)(131,142)(132,141)(133,140)(134,139)(135,138)(136,137)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,154)(152,153) );

G=PermutationGroup([(1,56,96,118,153),(2,57,65,119,154),(3,58,66,120,155),(4,59,67,121,156),(5,60,68,122,157),(6,61,69,123,158),(7,62,70,124,159),(8,63,71,125,160),(9,64,72,126,129),(10,33,73,127,130),(11,34,74,128,131),(12,35,75,97,132),(13,36,76,98,133),(14,37,77,99,134),(15,38,78,100,135),(16,39,79,101,136),(17,40,80,102,137),(18,41,81,103,138),(19,42,82,104,139),(20,43,83,105,140),(21,44,84,106,141),(22,45,85,107,142),(23,46,86,108,143),(24,47,87,109,144),(25,48,88,110,145),(26,49,89,111,146),(27,50,90,112,147),(28,51,91,113,148),(29,52,92,114,149),(30,53,93,115,150),(31,54,94,116,151),(32,55,95,117,152)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,88),(72,87),(73,86),(74,85),(75,84),(76,83),(77,82),(78,81),(79,80),(95,96),(97,106),(98,105),(99,104),(100,103),(101,102),(107,128),(108,127),(109,126),(110,125),(111,124),(112,123),(113,122),(114,121),(115,120),(116,119),(117,118),(129,144),(130,143),(131,142),(132,141),(133,140),(134,139),(135,138),(136,137),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,154),(152,153)])

95 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E ··· 10L 16A 16B 16C 16D 20A 20B 20C 20D 32A ··· 32H 40A ··· 40H 80A ··· 80P 160A ··· 160AF order 1 2 2 2 4 5 5 5 5 8 8 10 10 10 10 10 ··· 10 16 16 16 16 20 20 20 20 32 ··· 32 40 ··· 40 80 ··· 80 160 ··· 160 size 1 1 16 16 2 1 1 1 1 2 2 1 1 1 1 16 ··· 16 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

95 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 D4 D8 D16 C5×D4 D32 C5×D8 C5×D16 C5×D32 kernel C5×D32 C160 C5×D16 D32 C32 D16 C40 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 2 4 4 8 1 2 4 4 8 8 16 32

Matrix representation of C5×D32 in GL2(𝔽31) generated by

 2 0 0 2
,
 0 27 8 27
,
 27 29 23 4
G:=sub<GL(2,GF(31))| [2,0,0,2],[0,8,27,27],[27,23,29,4] >;

C5×D32 in GAP, Magma, Sage, TeX

C_5\times D_{32}
% in TeX

G:=Group("C5xD32");
// GroupNames label

G:=SmallGroup(320,176);
// by ID

G=gap.SmallGroup(320,176);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,309,1683,850,192,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c|a^5=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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